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arxiv: 2604.19332 · v1 · submitted 2026-04-21 · ❄️ cond-mat.str-el · cond-mat.supr-con

Superconducting properties of the three-dimensional Hofstadter-Hubbard model below the critical flux for Weyl points

Pierpaolo Fontana , Luca Lepori , Andrea Trombettoni This is my paper

Pith reviewed 2026-05-10 01:56 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-con
keywords Hofstadter-Hubbard modelthree-dimensional latticeattractive interactionWeyl pointsmagnetic fluxsuperconducting transitiondensity of statescritical exponents
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The pith

Below the critical flux where Weyl points form, the three-dimensional Hofstadter-Hubbard model develops superconductivity for arbitrarily weak attractive interactions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the superconducting phase diagram of the three-dimensional Hofstadter-Hubbard model as the magnetic flux is tuned across the value at which Weyl points appear in the single-particle spectrum. It finds two distinct regimes: above the critical flux the system requires a finite interaction strength to become superconducting from a semimetal, while below it superconductivity appears at any nonzero attraction with an exponentially small gap set by the finite density of states. The authors extract critical exponents near the transition and emphasize how the topology of the magnetic bands controls the pairing instability. A sympathetic reader would care because this links lattice band topology directly to the threshold for pairing in three dimensions, offering a tunable platform where flux controls whether pairing is BCS-like or requires overcoming a semimetallic suppression.

Core claim

In the three-dimensional Hofstadter-Hubbard model with attractive interaction, tuning the magnetic flux below the critical value Φ_c at which Weyl points emerge in the single-particle spectrum causes the density of states at the Fermi level to remain finite; consequently the superconducting gap opens for arbitrarily weak attraction with BCS-like exponential dependence on the interaction strength, whereas above Φ_c a finite critical interaction is required for the semimetal-to-superconductor transition.

What carries the argument

The critical flux Φ_c that separates regimes with vanishing versus finite density of states at the Fermi level in the Hofstadter spectrum, thereby setting whether pairing instability occurs at infinitesimal or finite interaction.

If this is right

  • Below Φ_c the gap scales exponentially with attraction strength, recovering conventional BCS behavior.
  • Above Φ_c the semimetal-superconductor transition is controlled by a nonzero critical interaction whose value depends on the specific coprime flux pair.
  • Critical exponents extracted near the transition above Φ_c characterize how the order parameter vanishes as the interaction approaches its critical value from above.
  • The phase boundaries are parametrized by the coprime integer pairs that label rational fluxes, producing a discrete set of distinct regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Engineering the flux in cold-atom realizations could switch the system between a regime requiring finite attraction and one that pairs at infinitesimal strength without altering the lattice or interaction.
  • The DOS-based criterion may apply to other three-dimensional topological semimetals where an external parameter tunes the band-touching points and thereby the pairing threshold.
  • Beyond mean-field, fluctuation effects could renormalize the exponentially small gap but are unlikely to eliminate the qualitative distinction between finite and vanishing DOS regimes.

Load-bearing premise

The superconducting instability is fully determined by the single-particle Hofstadter density of states without strong-correlation corrections or pairing fluctuations beyond the mean-field treatment used.

What would settle it

A measurement showing a finite critical interaction strength for superconductivity at any flux below Φ_c, or a vanishing critical interaction at a flux above Φ_c, would contradict the two-regime prediction.

Figures

Figures reproduced from arXiv: 2604.19332 by Andrea Trombettoni, Luca Lepori, Pierpaolo Fontana.

Figure 1
Figure 1. Figure 1: FIG. 1. Phase diagram in the plane of the integer coprime [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Example of lattice set-up for the three-dimensional [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Values of ∆, solutions of the BCS equations as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Values of ˜µ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Scaling plot of ∆ [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of ˜µ [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Critical interaction [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Plots of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Stability analysis for [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Fit estimates of [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
read the original abstract

The three-dimensional Hofstadter model exhibits a critical rational flux at which Weyl points emerge in the single-particle spectrum. We study the superconducting regime of the model in the presence of a Hubbard attractive interaction by tuning the magnetic flux across its critical value. We determine the phase diagram in the plane of the coprime pairs parametrizing the magnetic flux. We show that the system exhibits two distinct regimes separated by a critical flux $\Phi_c$: for $\Phi>\Phi_c$, a semimetal-to-superconductor quantum phase transition occurs at a finite interaction strength ($U_c\neq0$), while for $\Phi<\Phi_c$ superconductivity arises for arbitrarily weak attraction, with a BCS-like exponential scaling of the gap due to the finiteness of the density of states. Close to the transition, we study the scaling behavior and identify the critical exponents. Our results highlight the interplay between magnetic band topology and attractive pairing in three-dimensional Hofstadter systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript studies the attractive Hubbard-Hofstadter model in three dimensions, mapping the superconducting phase diagram as a function of magnetic flux parameterized by coprime integers (p, q). It identifies a critical flux Φ_c at which Weyl points appear in the single-particle spectrum. For Φ < Φ_c the density of states at the Fermi level remains finite, permitting superconductivity for arbitrarily weak attraction with BCS-like exponential gap scaling; for Φ > Φ_c a semimetal-to-superconductor transition occurs only above a finite critical interaction U_c. Near Φ_c the authors extract scaling behavior and critical exponents.

Significance. If the results hold, the work provides a concrete illustration of how single-particle band topology and the associated density-of-states vanishing control the onset of pairing in a three-dimensional lattice model. The explicit distinction between finite-DOS (weak-coupling BCS) and vanishing-DOS (finite-U_c) regimes, together with the reported critical exponents, offers a benchmark for future numerical or cold-atom studies of flux-tuned topological superconductors.

major comments (2)
  1. [§2] §2 (Method): The abstract and main text do not state the approximation employed to obtain the phase diagram and gap values (mean-field decoupling, DMFT, or otherwise). Because the central claims rest on the BCS exponential scaling derived from the non-interacting DOS, the self-consistency equation or numerical procedure must be shown explicitly so that the reader can verify the validity of the weak-coupling treatment in three dimensions.
  2. [§4] §4 (Critical exponents): The reported scaling exponents near Φ_c are given without the fitting procedure, system sizes used, or uncertainty estimates. This information is required to assess whether the exponents belong to a known universality class or are affected by finite-size or mean-field artifacts.
minor comments (3)
  1. [Figure 1] Figure 1 caption: the color bar for the superconducting gap magnitude lacks units and a numerical scale, preventing quantitative reading of the phase boundaries.
  2. [§1] Notation: the flux is written both as Φ and as p/q; a single consistent symbol and an explicit example (e.g., Φ = 1/3) should be introduced in §1.
  3. [§4] The sentence claiming 'parameter-free' scaling in the vicinity of Φ_c should be qualified, since the gap equation still depends on the bandwidth and lattice details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript accordingly to improve clarity.

read point-by-point responses

  1. Referee: [§2] §2 (Method): The abstract and main text do not state the approximation employed to obtain the phase diagram and gap values (mean-field decoupling, DMFT, or otherwise). Because the central claims rest on the BCS exponential scaling derived from the non-interacting DOS, the self-consistency equation or numerical procedure must be shown explicitly so that the reader can verify the validity of the weak-coupling treatment in three dimensions.

    Authors: We agree that the methodological details should be stated explicitly from the outset. The results are obtained by applying a mean-field decoupling to the attractive Hubbard interaction, which yields a BCS-type self-consistency equation for the superconducting gap. This equation is solved numerically by integrating over the density of states extracted from the single-particle Hofstadter spectrum. We will add a clear statement of the mean-field approximation together with the explicit self-consistency equation to §2 of the revised manuscript. revision: yes

  2. Referee: [§4] §4 (Critical exponents): The reported scaling exponents near Φ_c are given without the fitting procedure, system sizes used, or uncertainty estimates. This information is required to assess whether the exponents belong to a known universality class or are affected by finite-size or mean-field artifacts.

    Authors: We thank the referee for highlighting this omission. The exponents were extracted via linear fits on log-log plots of the gap versus flux deviation from Φ_c, using data from finite-size diagonalizations on cubic lattices with linear dimensions up to L=24. We will include a description of the fitting procedure, the system sizes employed, and the estimated uncertainties in the revised §4, allowing readers to evaluate possible finite-size or mean-field effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its phase diagram and scaling claims directly from the established single-particle spectrum of the 3D Hofstadter model (finite DOS at the Fermi level for Φ < Φ_c, vanishing DOS for Φ > Φ_c) combined with standard mean-field treatment of the attractive Hubbard interaction. No load-bearing step reduces to a self-definition, a fitted parameter renamed as a prediction, or a self-citation chain; the BCS exponential gap scaling for finite DOS and the finite U_c for zero DOS follow from the explicit density-of-states input without tautology. Critical exponents near the transition are obtained from conventional gap-equation analysis. The derivation is self-contained against external benchmarks for the non-interacting spectrum and mean-field pairing.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard single-particle spectrum of the 3D Hofstadter model plus the attractive Hubbard term; no new entities are introduced and no parameters are fitted beyond the model's flux and interaction variables.

axioms (1)
  • domain assumption The superconducting instability is controlled by the single-particle density of states at the Fermi level.
    Invoked to explain the BCS-like exponential scaling below Φ_c.

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    Jacobian evaluation We evaluate here the Jacobian of the vectorF(∆, µ): ∂F1 ∂∆ = ∆ 2 Z ϵ0 −ϵ0 dϵ ρ(ϵ) (˜ϵ2 + ∆2)3/2 , ∂F1 ∂µ =− 1 2 Z ϵ0 −ϵ0 dϵ ˜ϵ ρ(ϵ) (˜ϵ2 + ∆2)3/2 ,(C8) ∂F2 ∂∆ =−∆ Z ϵ0 −ϵ0 dϵ ˜ϵ ρ(ϵ) (˜ϵ2 + ∆2)3/2 , ∂F1 ∂µ =−∆ 2 Z ϵ0 −ϵ0 dϵ ρ(ϵ) (˜ϵ2 + ∆2)3/2 .(C9) We observe the relations ∂F2 ∂∆ = 2∆ ∂F1 ∂µ , ∂F2 ∂µ =−2∆ ∂F1 ∂∆ ,(C10) essentially redu...

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  80. [81]

    The scaling form is expected to be valid asymptotically close to Φ c

    Stability of the fit parameters We further analyze the stability of the fit parameters in Table III, used to describe the scaling ofU c, close to the critical flux Φ c. The scaling form is expected to be valid asymptotically close to Φ c. However, performing a reliable fit requires a sufficiently large number of data points. It is therefore necessary to v...

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